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Video Transcript

TranscriptBox-and-whisker plots

Deep in the mountains lies a martial arts school at 1 Foot Fist Way that focuses on breaking wooden planks. All of the students must break as many wooden planks as they can in one strike. Each student records his number and presents it to his master at the end of the week. The most consistently good student does not have to clean the school for one week. But how can the students' master tell which of his students are most consistently the best? By using Box-and-Whisker Plots, of course!

Using Box-and-Whisker Plots

If we want to put the students' data into a box-and-whisker plot, we need to have the numbers in order. Looking at student 1's record for the week, he has yet to order his list. After ordering the lists, we need to find 5 critical values: the minimum, the first quartile (also known as Q1), the median, the third quartile (also known as Q3), and the maximum.

Student 1

Heeding his teacher's instructions, student 1 orders his list. The minimum, or the smallest number, for student 1 is 1.
Student 1's maximum, or the largest number, is 9.
Next, let's find the median. The median is the middle number in the data set. Because we have an even number of data points, there are two middle numbers. When this happens, you should take the average of the two middle numbers. In this case, the average of 4 and 4 is 4. So now we know the minimum is 1, the median is 4 and the maximum is 9. To find each quartile, we must split the data into halves.

Q1 is the median of the first half of the data 1, 2, 2, 3, 4. The middle number of this portion of the data is 2, so the Q1 is 2.
Q3 is the median of the second half of the data 4, 6, 7, 8, 9. The middle number is 7, so the Q3 is 7.

Now that we have all 5 values, we can draw the box-and-whisker plot on our number line. Always plot the minimum, Q1, median, Q3, and maximum values The box part of the box-and-whisker plot is drawn with a vertical line through both the Q1 and Q3 values. These are then connected to form our box.

Finally, we also need to draw a vertical line in the box to represent the median. The Interquartile Range orIQR is obtained by substracting Q1 from Q3. In the case of student 1 this is 7 minus 5 or 2. The whiskers are then drawn to connect the box to the minimum and maximum values.

Student 2

Now let's look at student 2. First let's put the numbers in order. Now we need to find the 5 critical points again. Here, the minimum is 1 and the maximum is 8. Now let's find the median. Again, we have an even number of data points, this means we will have two middle values. The two middle values are 5 and 5, which when averaged, gives us 5. Now, we can split the data into halves in order to find Q1 and Q3.

The first half of the data is 1, 1, 2, 3, 5. So the Q1 is 2 because 2 is the median of the first half of the data. The second half of the data is 5, 6, 6, 8, 8. 6 is the median value in this portion of the data, so Q3 is 6.

Now that we have our five points, we can make a box-and-whisker plot. We draw a box from Q1 to Q3.Then, we make the whiskers by drawing lines from each end of the box to connect the minimum and maximum values.

Student 3

Let's put the data from student 3 in order. This diligent disciple has already completed his box-and-whisker plot! Let's check to see if all the parts are there. The minimum is 0, the maximum is 9... This time, even though the two middle numbers are different, we still just need to take the average. So, our median is the average of 3 and 4, or 3.5. Q1 is 1, Q3 is 7.

Comparing graphs

Points are plotted, box and whiskers drawn. Now we can compare the graphs and figure out which student is the most consistent. All three of these box-and-whisker plots are pretty similar, but they do have a couple of differences.The box part of the plot for student 2 is the shortest. This means that his data points are closer together; another way to say this is that student 2 has less variation in his data.

You may also notice that some of the critical values are different between the three plots. One critical point that varies the most between the three graphs is the median. Student 1 has a median of 4, student 2 has a median of 5, and student 3 has a median of 3.5. So even though student 1 and student 3 have the greatest maximums at 9, their medians are smaller than student 2.

Finally, the IQR values will show the teacher how consistent each student was. Student 1 has an IQR of 5, student 2 has an IQR of 4 and student 3's IQR is 6. So it's confirmed that student 2 is the most consistent. Before the teacher gets around to announcing the best student, the students clamor for him to show them how it's really done. Ahem. Deep in the mountains lies a martial arts school at 1/3 Foot Fist Way that focuses on breaking wooden planks, roads, trees, mountains.

About this Video Lesson

Description

A box-and-whisker plot is a quick way of showing the variability of a data set. It displays the range and distribution of data along the number line.

To make a box-and-whisker plot, start by ordering the data from least to greatest. Next, inspect the ordered data set to determine these 5 critical values: minimum, Q1, median, Q3, and maximum and plot them above a number line.

The minimum and maximum values are the least and greatest values. The median or middle value splits the set of data into two equal numbered groups. The first quartile, Q1, is the median of the lower half of the data set. The third quartile, Q3, is the median of the upper half of the data set.

The box is created by drawing vertical line segments through Q1, median, and Q3 and drawing two horizontal line segments connecting the endpoints from Q1 to Q3 passing through the median. The first whisker is created by drawing a horizontal line connecting the minimum and Q1 while the second whisker is created by drawing a horizontal line connecting Q3 with the maximum.

A good measure of the spread of data is the interquartile range (IQR) or the difference between Q3 and Q1. This gives us the width of the box, as well. A small width means more consistent data values since it indicates less variation in the data or that data values are closer together.